Judy and Steve play a game using five cards numbered 3, 4, 5, 8, 9. Judy chooses a card at random, looks at the number on it and replaces the card. Then Steve chooses a card at random, looks at the number on it and replaces the card. If their two numbers are equal the score is 0. Otherwise, the smaller number is subtracted from the larger number to give the score.

a) Show that the probability that the score is 6 is 0.08.

b) Draw up a probability distribution table for the score.

c) Calculate the mean score.

If the score is 0 they play again. If the score is 4 or more Judy wins. Otherwise Steve wins. They continue playing until one of the players wins.

d) Find the probability that Judy wins with the second choice of cards.

e) Find an expression for the probability that Judy wins with the nth choice of cards.